Thursday, August 27, 2015

Is China's turmoil the next Lehman Brothers?

I am taking a break from physics topics to discuss a hot topic of the day: the extreme turmoil on the stock markets due to trouble in China. I am not sure how the system works in other countries, but in US your retirement funds, your nest egg, is usually tied to the stock market fortunes as the largest investors are mutual funds. Every person have his or her tolerance to fluctuations and loss but major evaporation of retirement funds can be very nerve racking. Since I got burned in the past and learned from my mistakes, I want to share some common sense advice you will not easily find otherwise. 

The first observation is that unless you are a serious investor willing to commit time and resources into researching individual stocks it makes little difference overall on which mutual fund you select. Each mutual fund is already diversified and by watching how the DOW does you have a pretty good idea how your investments are doing. For the usual ups and downs of the market you do not need to worry at all, you will have an average return of about 8%, but what to do when there are violent price movements like the recent ones?

One of the most imbecile advice typically found on CNN by their so-called experts like Richard Quest is that you should not panic and ride out the storm. Is there a topic he is not qualified in? But when the losses exceed 8, 9, 10% this shows that major events are happening and it is not the time to stick your head in the sand and pretend it is business as usual. It is true that overreacting is bad and usually the ordinary investor crowd bets incorrectly. So you need expert advice to understand what is going on, but where do you get such a thing? One possible source is o reach out to your financial adviser, but in general this is not a good idea because those guys make money when you buy stocks and they do not offer unbiased advice. They usually give you the same cookie cutter nonsense of investing for the long term. So you have to realize that you cannot get good financial advice during an unfolding crisis and you should follow the money instead. 

If you watch the DOW value, get in the habit to see watch the transaction volume as well. The idea is that large financial institutions do have very serious experts who make decent predictions, and more important, those advice is actually followed. And when large mutual funds decide to make a large move, the volume tells the story. So the advice is simple:
  • when DOW drops significantly but the volume is close to the average, don't panic and do nothing.
  • when DOW drops significantly and the volume is two to three times the average it is time to sell. The large players decided that the outlook is negative. 
  • when DOW increases significantly but the volume is close to the average, do nothing.
  • when DOW increases significantly and the volume is two to three times the average it is time to buy. The large players decided that the outlook is positive. 
Sure, you will miss the first big drop, and the first large increase, but in the long term this does not matter. This advice is for the investors who don't have the stomach for large drops in value. If your tolerance for risk is higher or you have the luxury of a really long term investment period measured in many decades then you should not care about the stock market antics in the first place.   

Now on the current China turmoil. 

Is this a short storm, or the harbinger of a larger trouble like the Lehman Brothers collapse? Honestly, I don't know, and I think nobody knows either. China is now the second largest economy and any turmoil there has large implications. Also the current trouble is the result of irresponsible advice from Chinese leadership to ordinary citizens to buy stocks in a get rich quick scheme. This created a bubble which now burst. But what will the effect be in the psychology of ordinary Chinese citizens? Would this create unbearable social pressure which will result in the change of the political system? Would this impact the real Chinese economy? Nobody knows because this is a first in the modern history of China. When Lehman collapsed, for a few months there was no economic pain. Everyone was reassured that "the economic fundamentals are strong" and this was just a blimp on the radar. But something funny happened: the usual economic activity simply stopped like a switch was turned off because lending between banks froze solid due to sudden lack of confidence. It took a Keynes style influx of money to get the economic engine restarted.

Thursday, August 20, 2015

Where does the Hilbert space come from?

Continuing the discussion from last time, today we can put some of the pieces of the puzzle together. It is helpful to switch the discussion direction from classical to quantum and start for a moment from the quantum mechanics side to see where we want to arrive.

Why do we use a Hilbert space in quantum mechanics? This is a big topic and we cannot cover it in one (or even several posts). Right away we will restrict ourselves to the finite dimensional degrees of freedom case, thus excluding the field theory considerations and avoiding the issues raised by Haag's theorem, or the breakdown of the Stone-von Neumann uniqueness theorem. We will also skip the treatment of unbounded operators which require the theory of rigged Hilbert spaces and we will stick with boring but well behaved bounded operators. 

For a bounded operator T on a Hilbert space it is easy to prove that \(||T^{\dagger} T|| = {||T||}^2\) as follows:

\({||T \Phi||}^2 = \langle T \Phi, T \Phi\rangle = \langle T^{\dagger} T \Phi, \Phi\rangle \leq ||T^{\dagger} T\Phi || ||\Phi|| \leq ||T^{\dagger} T || {||\Phi||}^2 \)
\({||T||}^2 \leq ||T^{\dagger} T || \)
and since 
\(||T^{\dagger} T || \leq ||T^{\dagger}|| ||T|| = {||T||}^2\)
we have:
\({||T||}^2 \leq ||T^{\dagger} T || \leq {||T||}^2\)

An algebra of bounded operators on a Hilbert space is the prototypical example of a C* algebra. A remarkable fact is the correspondence between states and representations of C* algebra given by the GNS construction. Here a representation is a linear map from the elements of the C* algebra to bounded operators on some Hilbert space.

From categorical considerations one can obtain a C* algebra without the norm axioms. To distinguish math from physics one needs to be able to make experimental predictions and this is where the states enter the picture. A state on a C* algebra gives rise to a representation of the algebra as bounded linear operators on some Hilbert space and this is how Hilbert spaces are introduced. The key ingredient for this to work is the C* norm condition: \(||T^{\dagger} T|| = {||T||}^2\). However, this norm is unique and is given by the spectral radius - an algebraic concept! So there is hope we can arrive at quantum mechanics using only algebraic methods. Now we will show how.

Coming back to the quantization discussion from the prior post, what we need to achieve is a prescription which constructs operators on a Hilbert space from functions on the phase space (also known as the cotangent bundle). Even better we should be able to start from either a Kahler, symplectic, or Poisson manifold.

We can start with the simplest case where we replace the position \(q\) and the momenta \(p\) with the operators: \(x\) and \(\frac{h}{i}\frac{\partial}{\partial x}\) in any observable \(f(p, q)\) provided \(f\) contains no products \(pq\) because the position and momenta operators in the Hilbert space do not commute and the order of the operators is ambiguous.

The next level of sophistication is Weyl quantization procedure and the details can be found here. (Please excuse me for skipping typesetting it in LaTeX.) Weyl quantization tends to preserve well symmetry properties, but a better quantization prescription is Berezin quantization which work on all Kahler manifolds when positivity is guaranteed by the Kodaira embedding theorem.

Erich Kahler

A Kahler manifold is a truly outstanding mathematical object where three concepts meet:

  • a metric structure
  • a symplectic structure
  • a complex structure
and any two define the third one. The main example is the complex projective space (endowed with the Fubini-Study metric) which is essential for quantum mechanics. It is very enlightening to see how it all works out in quantum mechanics and I'll attempt to show it below.

In classical and quantum mechanics there are two products, one symmetric \(\sigma\) and one anti-symmetric \(\alpha\) corresponding to observables and generators as follows:

Observables: \(\sigma\) = regular function multiplication on phase space OR Jordan product
Generators: \(\alpha\) = Poisson bracket OR commutator

There is also a 1-to-1 map \(J\) between observable and generators called dynamic correspondence where \(J^2 = 0\) for classical mechanics and \(J^2 = -1\) for quantum mechanics. This map corresponds to Noether's theorem.

Composing two physical systems 1 and 2 gives rise to the following fundamental composition relationship:

\(\sigma_{12} = \sigma_1 \otimes \sigma_2 + J^2 \frac{\hbar^2}{4}\alpha_1 \otimes \alpha_2\)
\(\alpha_{12} = \sigma_1 \otimes \alpha_2 + \alpha_1 \otimes \sigma_2\)

and so the symmetry and anti-symmetry of the products is preserved.

Now we want to deform the Poisson bracket and regular function multiplication of classical mechanics which respects the composition with \(J^2 = 0\) into two products which respect \(J^2 = -1\). We can do this term by term in powers of \(\hbar\) preserving associativity at each step. This is the essence of deformation quantization.

Without ado, here is the solution given by Moyal sine and cosine brackets in terms of the Poisson bracket \(\{ , \}\) in the simplest case of a flat space:

\(\alpha = \frac{2}{\hbar} sin (\frac{\hbar}{2} \{ , \})\)
\(\sigma = cos (\frac{\hbar}{2} \{ , \})\)

The star product is then \(\star= \sigma+ J\frac{\hbar}{2} \alpha\) and we arrived at quantum mechanics in phase space.

First a note: I demanded earlier to preserve associativity at each power of \(\hbar\). This is a physical requirement to be able to compose experiments sequentially and not care where we draw the boundaries between them. But this has a very interesting consequence: we have freedom of pick how we carry out the quantization at each power of \(\hbar\) step and this makes the subject of quantization non-trivial. In particular it turns out that the equivalence classes of star products on symplectic manifolds are in 1-to-1 correspondence with the second de Rham cohomology \(H^2_{dR} (M)\)!

Second, we can see where the inner product is coming from. From the Moyal sine bracket we extract a symplectic form \(\omega^{IJ}\) and construct it's inverse \(\Omega_{IJ}\). So we have one of the three structures of a Kahler manifold: the symplectic structure. But we also have the complex structure as well because we have \(J^2 = -1\). It can be shown that \(J\) is actually a tensor or rank (1,1): \(J = J^{I }_{  J}\) and from this we get a metric tensor \(g_{IJ}\):

\(g_{IJ} = \Omega_{IK} J^{K}_{  J}\)

The complex inner product is defined now by: \(g+ \sqrt{-1}\Omega\):

\(\langle X, Y \rangle = X^{T} g Y + i X^T \Omega Y \)

where X and Y are column vectors : \(q_1, q_2,...q_n, p_1, p_2, ..., p_n\)

Time evolution preserves \(J\) and \(\Omega\), meaning they preserve the metric structure by preserving a normalization constraint:

\( \langle g \rangle - 1=  X^I g_{IJ} X^J - 1 = 0\)

The constraint Hamiltonian motion which preserves the metric structure is nothing but the Schrodinger equation is disguise!

I do not want to create the impression that this is all as simple as this. I only discussed the flat \(R^{2n}\) case above. There are many subtle and hard problems, as well as open questions. As an example, there are Poisson manifolds which do not admit a Kahler structure, but all Poisson manifolds are quantizable. How would the quantization of such a system look like? Perhaps there are no bounded operators in this case, I don't know.

Next time I'll present a concrete calculation of a standard problem in the phase space formalism of quantum mechanics. This will challenge the ontic interpretation of quantum mechanics.

Friday, August 14, 2015

Cotangent bundles, symplectic, and Poisson manifolds

For today's post I want to explore some interesting facts about classical mechanics. But why should we care about classical mechanics if nature is quantum at core? One misconception I used to have was that we should care only about the emergence of classical physics out of quantum mechanics. I was not alone in this misconception, the late Sidney Coleman supported this point of view in his outstanding talk: Quantum Mechanics in your face. But as it turns out rigorous deformation quantization starting from a Poisson manifold can help arrive at the standard formalism of quantum mechanics. 

And to achieve that we need to understand the mathematical structures involved in classical mechanics: cotangent bundles, symplectic manifolds, Poisson manifolds. Because I did not write the needed mathematical posts for symplectic geometry I will keep the discussion at high level.

The natural starting point for classical mechanics is a configuration space Q of generalized coordinates and Hamiltonian mechanics happen in the cotangent bundle \(T^* Q\): to each point of Q we attach the dual of the tangent space. The transformation between the velocities in the tangent bundle and the momenta in the cotangent bundle is made by the Lagrangian L using a Legendre transform:

\(p_i = \frac{\partial L}{\partial v_i}\)

The Hamiltonian is obtained from the Lagrangian using a Legendre transformation:

\(H(q^i, p_j) = p_i v^j - L(q^i, v^j)\)

While the Lagrangian formulation is best suited for field theory and its relativistic formulation, the Hamiltonian formalism is more natural for classical and quantum mechanics. One reason to use the cotangent bundle over the tangent bundle is the existence of a canonical "God given" 1-form:

\(\theta = \sum_{i} p_i dq^i\)

The hierarchy of spaces used in classical mechanics is:

cotangent bundle \(\subset\) simplectic manifold \(\subset\) Poisson manifold

To go from the cotangent bundle to a symplectic manifold we use the canonical 1-form \(\theta\):

\(\omega = d \theta\)

A manifold with a closed nondegenerate differential 2-form \(\omega\) is called a symplectic manifold. Not all symplectic manifolds are cotangent bundles, but due to Darboux theorem a symplectic manifold is locally a cotangent bundle. In a symplectic manifold a polarization is a Lagrangian foliation (in the tangent bundle the foliation is the map \((q, p) \rightarrow q\); picture a foliation as a slicing of the space). Existence of polarizations is essential for proving the equivalence of a symplectic manifold with a cotangent bundle when true. Moreover polarizations are key ingredients in geometric quantization. The details are much more complex but this is the 10,000 feet view.

The generalization from symplectic to Poisson manifolds happen when \(\omega\) can be degenerate. Non-degeneracy demands that the dimension of a symplectic manifold to be even, but Poisson manifolds can have odd dimensions. So how does the Poisson bracket look on a Poisson manifold?

The prototypical is the free pivoted rigid body rotation. The Hamiltonian is:

\(H(x) = \frac{1}{2}(\frac{x^2}{I_x} + \frac{y^2}{I_y} + \frac{z^2}{I_z})\)

the Poisson bracket is:

\(\{F,G\}(x) =-x \cdot (\nabla F \times \nabla G)\)

which makes the Hamiltonian equations of motion:

\(\frac{d x}{d t} = x\times \nabla H(x)\)

The most general Poisson bracket can be proven to be:

\(\{F, G\} = \sum_{i,j} \{x_i , x_j\} \frac{\partial F}{\partial x_i} \frac{\partial G}{\partial x_j}\)

and again by Darboux theorem for any Poisson manifold there are local coordinates for which the Poisson bracket takes the usual form for the even subset of coordinates for which the rank is locally constant. In other words, a Poisson manifold can be decomposed into a product of a symplectic manifold and a Poisson manifold of rank zero.

But how do we get the crazy Poisson bracket from above? The answer is symplectic reduction. I did not built the mathematical pre-requisites of moment maps to explain in an intuitive way how this works, and this will have to wait for future posts. In the meantime I can state some conclusions.

First, category arguments can reconstruct both classical and quantum mechanics. In the classical case those arguments offer no help in proving the existence and non-degeneracy of \(\omega\). All we get from category arguments are Poisson manifolds! So it is imperative to prove that one can use deformation quantization on any Poisson manifold to arrive at quantum mechanics. Fortunatelly such a proof exists and was found by Maxim Kontsevich in 1997: A remarkable fact is that the proof uses string theory insights!!!

Second, the existence of time without invariance under composition is not enough to reconstruct either classical or quantum mechanics. The best counterexample comes from soliton theory and the Korteweg-de Vries equation. A KdV equation comes from a Poisson manifold without a symplectic analog, and moreover it is a bi-Hamiltonian system. KdV admits not one, but two Poisson brackets compatible with each other!

Third, not all quantum systems come from quantization of classical systems.

Next time I'll start exploring topics of quantization. Please stay tuned.

Friday, August 7, 2015

From time to quantum mechanics

Last time I presented the case for canonical time evolution stemming from the non-commutativity of operator algebras, and today I'll start talking about the reverse implication: obtaining quantum mechanics from the existence of time. 

First, let me expand on the brief statement from the prior post that the transition from classical to quantum mechanics is NOT as simple as replacing the Poisson bracket with the commutator. We need to put this in rigorous mathematical formalism and this is known as the Dirac problem:

"Does  there exists two matrices P and Q and a correspondence \(\phi\) which, to every polynomial g in the classical variables p and q, associates a matrix \(\phi (g)\) in such a manner that:
(i) \(\phi (p) = P\) and \(\phi (q) = Q\)
(ii) to the unit function \(1 : (p, q) \in R^2 \rightarrow 1\), \(\phi\) associates the unit matrix I
(iii) \(\phi\) is linear
(iv) \((i/\hbar) [\phi (f), \phi(g)] = \phi(-\{f,g\})\) for every polynomials f and g in p and q
(v) the matrices P and Q form an irreducible system, i.e. the only matrices A satisfying [X,P] = 0 = [X,Q] are of the form \(X = \lambda I\) where \(\lambda \in C\) and I is the unit matrix?"

It turns out that the Dirac problem has no solution and the proof is actually not very complicated. What are really complicated are the solutions to the quantization problem, and different approaches reject different assumptions in Dirac's problem. 

What is important for our purposes is that in classical mechanics one encounters the Poisson bracket and the quantum mechanics we have the commutator. Moreover both of them obey Leibniz identity:

\(\{H, fg\} = \{H, f\} g + f \{H, g\}\)
\([H, AB] = [H,A] B + A [H, B]\)

In the case of the Poisson bracket this is a trivial consequence of the partial differential operators, while in the case of the commutator this is a simple algebraic identity:

\([H, AB] = HAB-ABH = HAB-AHB + AHB -ABH = [H,A] B + A [H, B]\)

We can understand the commutator and the Poisson bracket as a product "\(\alpha\)":

\(A \alpha B = [A, B]\)
\(f \alpha g = \{f,g\}\)

and this is related to time evolution. If we call T a time translation operator and o any algebraic product used in physics, the invariance of the laws of Nature under time evolution implies the following commutative diagram:

T(A) o T(B) = T(A o B) which shows that [T, o] = 0 (T after o is the same as o after T) or that time translation preserves algebraic relations.

In the infinitesimal case in natural units (ignoring the usual factors of h bar and such): \(T = I + \epsilon H \alpha\) 

Substitution in T(A) o T(B) = T(A o B) yields:

\(((I + \epsilon H \alpha) A)  o ((I + \epsilon H \alpha) B) = ((I + \epsilon H \alpha )(A o B))\)

which to first order in \(\epsilon\) is:

\(H \alpha (A o B) = (H \alpha A) o B + A o (H \alpha B) \)

which is known as Leibniz identity. (A trivial observation is that when \(o = \alpha\), the Leibniz identity becomes the Jacobi identity and this gives rise to a Lie algebra.)

Now the heavy mathematical lifting follows using category theory arguments (invariance under composition = universality of the theory) to completely recover the algebraic structure of quantum and classical mechanics. Then using geometric or deformation quantization (to avoid the lack of the solutions for Dirac's problem) one obtains the usual Hilbert space formalism for quantum mechanics. Therefore the Poisson bracket and commutator are the only mathematical realizations of Leibniz identity for theories of nature obeying invariance under composition.

The starting point of quantum mechanics reconstruction using category theory arguments is Leibniz identity and this follows from infinitesimal time translations of the commutative diagram from above.

Moreover, in the usual Hilbert space formulation Leibniz identity corresponds to unitarity. This cuts both ways and violation of unitarity implies violation of Leibniz identity. And so now we have a big mathematical problem:

collapse postulate -> unitarity violation -> violation of Leibniz identity -> no Hilbert space formalism for quantum mechanics!!! 

It makes no more sense to talk about Hilbert space, operators, etc and this is clearly impossible. Something must give. Is it that there is no real collapse (MWI)? Or do we need to add contextual protection (Bohmian, QBism)? Or maybe there is an extension of quantum mechanics (GRW)?

How can this be solved? The so-called measurement problem just got much more serious than a simple problem of philosophical interpretation. There is good news however: the same categorical arguments which highlighted the problem in the first place, point the way to a most natural solution: unitary dynamical generation of superselection rules similar with spontaneous symmetry breaking. To be continued...